[aliromman]

This scoped negative result closes two fixed-source BIT price questions. Adding fresh commuting axes to any nonempty helper set in the retained PR589 word costs more than the added target credit for every positive axis count and every tariff r^(1-C), 0<C<1. The retained paired-triple p12 invoice, with both main boundaries and all helper credits, has a raw saving below 0.002 for every positive integer parameter in the balanced query-thinned caller. The complete kappa calculation does not improve.

The proof, exact result, and source pins give the two separate scopes. The base and every live dependency are pinned at 32890bc5de034ab8fc93ac88ead7279a6a416d04. No source, selected result, or other contributor's package changes.

For the first gate, the actual 439 helper jobs have rank eight, while every MOVE on a helper's own paid path has rank at most seven. A new axis must cross such a path. Strict concavity and subadditivity make its paid increment exceed its target increment. The proof allows partial axis sets and different positive axis counts on different helpers. It fixes the source word, scalar contacts, free maps on the new axes, and other target credits. It does not cover a changed word, preloaded axes, status-changing free maps, shared persistent ownership, or a different output contract.

For the second gate, the geometry parameter is fixed at 12: h=24, v=1760, R=19788, and centre debt 528. The retained generator reconstructs the complete invoice from frozen data. Exact rational checks give negative margin at C=1/500 for caller parameters 1 through 64. A concave upper bound has negative value and derivative at 64, which closes the remaining tail. A separate normalized-margin argument bounds each raw crossing below 1/500. The sign-change bracket (0.00164539220, 0.00164539221) at caller parameter 11 is a control, not a global optimum claim. The retained rank compiler, its documented modular-rank argument, and its native invoice convention are inherited. This gate does not prove a new native admission or cover another geometry, partial-gauge change, local recipe, or caller.

A separate appendix proves that min rank(J+P)=9 over all permutation matrices P, for the pinned 111-by-111 PR589 noCOPY matrix J. Its exact 511 nonzero defect-code words have minimum weight 48. A putative rank-eight repair could move at most 34 rows, which cannot remove the rank-nine defect. The identity attains nine. This is a binary correction-rank result, not a native-factorization-cost bound. The retained rank-three arbitrary-invertible completion has a different output class and does not contradict it.

Use Python 3.12 and the standard library:

python3 -B research/bit-fixed-source-price-gates/verify.py
python3 -O -B research/bit-fixed-source-price-gates/verify.py
python3 -B scripts/research_package.py --package research/bit-fixed-source-price-gates --check-only
python3 -B scripts/research_package.py --package research/bit-fixed-source-price-gates

Expected direct output:

PASS_BIT_FIXED_SOURCE_PRICE_GATES helper_axis=all_k_C paired_k3=all_controller_p source_controls=10 new_kappa=false

Normal and optimized exact reports match. The registered Python 3.12 replay, 17-file committed-byte and export checks, and normal/optimized replay in a fresh Git-free export passed. The verifier checks all ten source pins before loading retained code, rejects a mutation of each input, and rejects five invalid price assumptions and two invalid defect-code controls. It checks the two helper rank censuses, complete paired-triple invoice, 64 exact margins, analytic tail, positive crossing fixture, and all 511 nonzero codewords.

The source word comes from PR #589 and PR #555. The source notice credits eumemic, substantial Anthropic Claude assistance, and earlier contributors. The fast paired-cube generator's source record pins the PR #168 lineage at 4a3c769e5c5430e7114c4d3e099ff34664677f17 and the frozen p12 data; its frame and gauge ledger retains PR #144 credit. The exact series engine comes from ai-nanachi's compact productive BIT package. Its credit record identifies Cinna Davis's query-thinned controller lineage. Source-bound proofs, checks, and packaging have substantial OpenAI Codex assistance. New files are Apache-2.0; retained notices are unchanged.

Related contracts remain distinct: PR #1931 changes the E8 source with boundary repairs; PR #1998 decodes ordered paired dirty logs; PR #2376 maps QQ16 observation contracts. PR #673 and PR #1246 concern priced dirty restoration and reader-defect repair, not the fixed matrix-to-permutation rank problem. These comparisons are not executable dependencies.

No mincut, matching, floating-root, or new-geometry discovery search was rerun. The predecessor's standalone independent check_paired_cube_bit.py was not rerun; the second gate prices the reconstructed retained generator invoice. No native payload, complete cover, ordinary precision or movement budget, uniform compiler, or outer multiplication assembly was checked. The optional full-history guard is unavailable in this shallow checkout. Hosted CI has not run at publication. The next questions require a concrete changed word and every target debit before another price claim.

A fresh overlap check also found PR #5512, which improves the ordinary BIT component through a changed G20 word and owned relations. This package fixes the earlier sources and does not bound that changed contract. PR #5465 separately applies concavity to completion of the complex source helpers; the first gate here adds fresh commuting axes on the fixed BIT word.